| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s − 2·13-s − 2·15-s + 4·17-s − 19-s + 21-s + 6·23-s − 25-s + 27-s − 6·29-s + 5·31-s − 2·35-s − 7·37-s − 2·39-s + 4·43-s − 2·45-s − 4·47-s − 6·49-s + 4·51-s − 6·53-s − 57-s − 6·59-s − 61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 0.970·17-s − 0.229·19-s + 0.218·21-s + 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.898·31-s − 0.338·35-s − 1.15·37-s − 0.320·39-s + 0.609·43-s − 0.298·45-s − 0.583·47-s − 6/7·49-s + 0.560·51-s − 0.824·53-s − 0.132·57-s − 0.781·59-s − 0.128·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.57459684772903, −15.19675317929635, −14.70587148797562, −14.19101457067878, −13.72685946139698, −12.90083474195355, −12.53513909459987, −11.98708546199670, −11.32435536349971, −10.97235645374613, −10.18325875026010, −9.626700330848630, −9.097231591499507, −8.358837381649112, −7.945049483186999, −7.458936958019496, −6.924320919500410, −6.157580124990518, −5.213110937788825, −4.840540612894260, −4.010056551138696, −3.448362584732337, −2.850222527223412, −1.946609220398205, −1.115958259117707, 0,
1.115958259117707, 1.946609220398205, 2.850222527223412, 3.448362584732337, 4.010056551138696, 4.840540612894260, 5.213110937788825, 6.157580124990518, 6.924320919500410, 7.458936958019496, 7.945049483186999, 8.358837381649112, 9.097231591499507, 9.626700330848630, 10.18325875026010, 10.97235645374613, 11.32435536349971, 11.98708546199670, 12.53513909459987, 12.90083474195355, 13.72685946139698, 14.19101457067878, 14.70587148797562, 15.19675317929635, 15.57459684772903
